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Question # (square roots of the identity matrix) For how many 2x2 matrices A is it true that A^2=I ? Now answer the same question for n x n matrices where n>2

Matrices
ANSWERED (square roots of the identity matrix) For how many $$2\times2$$ matrices A is it true that $$A^2=I$$ ? Now answer the same question for $$n\times x$$ n matrices where $$n>2$$ 2021-03-09

Step 1
To find how many matricec A of order 2 exists such that $$A^2=I$$
Here we have to find $$A^2=A \times A =I$$
Let us consider any arbitrary matrix of order 2. $$A=\begin{pmatrix}a & b \\c & d \end{pmatrix}$$
Then $$A^2$$ is given by $$A^2=\begin{pmatrix}a & b \\c & d \end{pmatrix} \begin{pmatrix}a & b \\c & d \end{pmatrix} = \begin{pmatrix}a^2+bc & ab+bd \\ca+dc & d^2+bc \end{pmatrix}$$
Now from $$A^2=I$$ , we get $$A^2 = \begin{pmatrix}a^2+bc & b(a+d) \\c(a+d) & d^2+bc \end{pmatrix} = \begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix} =I$$
$$(1) a^2+bc=1$$
$$(2) d^2+bc=1$$
$$(3) b(a+d)=0$$
$$(3) c(a+d)=0$$
Step 2
Case 1:
If $$(a+d) \neq 0$$
From (3) we get : $$b=0$$
From (4) we get : $$c=0$$
From (1) we get : $$a^2=1 \Rightarrow a=\pm1$$
From (2) we get : $$d^2=1 \Rightarrow d=\pm1$$
Therefore we get the matrix of the form
$$\left\{A=\begin{pmatrix}a & 0 \\0 & d \end{pmatrix} : a=\pm1 , d= \pm1 \right\}$$ Therefore there are only 4 matrices exist such that $$A^2=I$$ from case 1.
Step 3
Case 2:
If $$(a+d)=0$$
sub case a:
If a=0 , that implies $$d=0$$
Then from(1 ) and (2) , we get : $$bc=1 \Rightarrow c=b^{-1}=\frac{1}{b}$$ in real numbers.
Therefore we get the matrix of the form
$$\left\{A=\begin{pmatrix} 0 & a \\a^{-1} & 0 \end{pmatrix} :a \in F \right\}$$ Therefore there are infinitely many matrices exist if the field is the set of real numbers such that $$A^2=I$$
Step 4
We can also consider further subcases to find the type of involuntary matrices, but as the question is asked to find how many, we have already got infinitely many $$2 \times 2$$ matrices of such type.
Similarly for any $$n>2$$ , we can find infinitely many matrices satisfying this condition on the field of real numbers.